Nonzero minimal entropy for four-manifolds with hyperbolic-product decompositions

Let $M$ be a smooth orientable four-manifold that admits a proper geometric decomposition into pieces modelled on ${\mathbb H}^2 \times {\mathbb H}^2$, and let ${\rm h}(M)$ denote its minimal entropy. The nonzero minimal entropy conjecture. If $M$ admits such a decomposition, then

$${\rm h}(M) \neq 0.$$

The preceding discussion establishes positive simplicial volume for geometrisable four-manifolds containing real or complex hyperbolic pieces and expects an analogous conclusion for decompositions with pieces modelled on ${\mathbb H}^2 \times {\mathbb H}^2$.